It's a bit confusing to equate the first appearances of characters with "starting". In most cases, the characters have had long careers before we first meet them, and the details of those careers are later revealed in part or in whole.
OTOH, we never learned much about Sulu's past, but apparently there was nothing relating to engineering there; all we saw was a stint at astrophysics.
Incidentally, we did see a non-com navigator take the place of Lieutenant Ilia in ST:TMP, giving some precedent to the idea of noncoms handling jobs where the overwhelming majority of performers are commissioned officers. Transporter control is possibly another such field, but it's difficult to tell for sure: in TOS, plenty of people wore officer braid in the role, but those red coveralls and the lack of Ensign braid confused the issue - and in TNG, officer pips were common for transporter chiefs including O'Brien, but in O'Brien's case these were supposed to be a costuming error in retrospect.